Boundary conditions

The temperature $\theta$ inside the geometrical object is driven by

  1. the material properties and the initial temperature and
  2. the heat flux along the geometrical boundary sides.

We denote the heat flux

  • inwards the object as induced heat flux $\Phi_{in}$ and
  • outwards the object as emitted heat flux $\Phi_{out}$ or Emission.

Mathematically spoken, the situation on the boundary sides are described here by Neumann or Robin boundary conditions. On boundary sides where actuators (e.g. heating elements) are assumed, we have

\[\left[\lambda(\theta) \nabla \theta(t,x) \right] ~ \vec{n} = \Phi_{in}(t,x) + \Phi_{out}(t,x)\]

with outer normal vector $\vec{n}$.

Emitted heat flux / Emissions

The emissions are described by

  • linear heat transfer / convection

\[-h (\theta - \theta_{amb})\]

  • and nonlinear heat radiation

\[- \epsilon ~ \sigma (\theta - \theta_{amb})\]

with ambient temperature (temperature of the object's surounding) $\theta_{amb}$, heat transfer coefficient $h>0$, emissivity $\epsilon \in [0,1]$ and Stefan-Boltzmann constant $\sigma$.

Heat transfer coefficient $h$ and emissivity $\epsilon$ can be defined for each boundary side seperately. Internally, the multiplication of emissivity $\epsilon$ and Stefan-Boltzmann constant $\sigma$ is saved as radiation coefficient $k = \epsilon ~ \sigma$.

Hestia.Emission — Type
Emission  <: AbstractEmission

Type Emission contains the coefficients for (convective) heat transfer and heat radiation

\[\Phi = -h ~ (\theta - \theta_{amb}) - \sigma ~ [\varepsilon_{1} ~ \theta^4 - \varepsilon_{2} ~ \theta_{amb}^4)]\]

Constructor Emission(h, θamb, ε₁, ε₂, θext) expects emissivity ε₁ and ε₂ which must be in interval [0,1]. The Stefan-Boltzmann constant is stored internally: σ = 5.6703744191844294e-8.

Elements

h : heat transfer coefficient

θamb : ambient temperature

ε₁ : Emissivity of main object

ε₂ : Emissivity of external second body

θext : temperature of an external second body

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Hestia.emit — Function
emit(temperature :: Real, emission :: Emission)

Calculates the right-hand side of the boundary conditions for a given Emission.

\[\Phi = -h ~ (\theta - \theta_{amb}) - \sigma ~ [\varepsilon_{1} ~ \theta^4 - \varepsilon_{2} ~ \theta_{amb}^4)]\]

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Hestia.emit! — Function
emit!(flux :: Vector{<:Real}, temperature :: Vector{<:Real}, emission :: Emission)

Calculates the right-hand side of the natural Robin boundary along a boundary for a given Emission.

\[\Phi = -h ~ (\theta - \theta_{amb}) - \sigma ~ [\varepsilon_{1} ~ \theta^4 - \varepsilon_{2} ~ \theta_{amb}^4)]\]

Note: in-place operation - results are saved in array flux.

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Induced heat flux

The induced heat flux is generated by $N_{u}$ actuators that can be placed directly (e.g. heating elements) or indirectly (e.g. laser beam) on each boundary of the geometrical object. All actuators are assumed to have a specific spatial characterization $b_{n}(x)$ reaching from 0 to 1 like a scaling from zero to 100%. The control signal $u_{n}(t)$ defines the power intensity of the n-th actuator. So, the induced heat flux is defined by

\[\Phi_{in}(t,x) = \sum_{n=1}^{N_{u}} b_{n}(x) ~ u_{n}(t).\]